Godzilla is a beast but has a very harsh ride, even in Comfort mode. The GTR transmission is great for daily driving (never mind the issues about breakage if you launch too hard/too often). Other than the stiff ride it's totally civilized. The Vette is also very civilized and isn't quite as punishing over rough roads, but the clutch can wear you out in traffic. The Vette gets much better gas mileage, if you care about that. Tall 6th gear FTW.
I think you should drive them both, they are radically different cars and your preference will probably make itself clear after you've experienced the differences between them.
As far as price goes, Z06 is the winner hands down. You can get an 08 for around $59k with the base options. Loaded out closer to $70k.
Be aware that with the GTR you are signing up for some very big maintenance bills. There is no free service (other than the very first post break-in service) and some of the costs are steep. A replacement set of pads and rotors will run you $6,000. I repeat, $6,000. Even an oil change will be a couple of hundred bucks or more because it requires removing the undertray.